In mathematics, Arakelyan's theorem is a generalization of
Mergelyan's theorem Mergelyan's theorem is a result from approximation by polynomials in complex analysis proved by the Armenian mathematician Sergei Mergelyan in 1951.
Statement
:Let ''K'' be a compact subset of the complex plane C such that C∖''K'' is conne ...
from compact subsets of an open subset of the complex plane to relatively closed subsets of an open subset.
Theorem
Let Ω be an open subset of
and ''E'' a relatively closed subset of Ω. By Ω
* is denoted the
Alexandroff compactification In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Ale ...
of Ω.
Arakelyan's theorem states that for every ''f'' continuous in ''E'' and holomorphic in the interior of ''E'' and for every ''ε'' > 0 there exists ''g'' holomorphic in Ω such that , ''g'' − ''f'', < ''ε'' on ''E'' if and only if Ω
* \ ''E'' is connected and locally connected.
See also
*
Runge's theorem
In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Runge who first proved it in the year 1885. It states the following:
Denoting by C the set of complex numbers, let ''K ...
*
Mergelyan's theorem Mergelyan's theorem is a result from approximation by polynomials in complex analysis proved by the Armenian mathematician Sergei Mergelyan in 1951.
Statement
:Let ''K'' be a compact subset of the complex plane C such that C∖''K'' is conne ...
References
*
*
* {{cite journal, last1=Rosay, first1=Jean-Pierre, last2=Rudin, first2=Walter, title=Arakelian's Approximation Theorem, journal=The American Mathematical Monthly, date=May 1989, volume=96, issue=5, pages=432, doi=10.2307/2325151, jstor=2325151
Theorems in complex analysis
Theorems in approximation theory